An explicit self-dual construction of complete cotorsion pairs in the relative context

Abstract

Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal{F},\mathcal{C})$ be a hereditary complete cotorsion pair in $R-\mathsf{Mod}$. Let $({\mathcal{F}}_A,{\mathcal{C}}_A)$ be the cotorsion pair in $A-\mathsf{Mod}$ in which ${\mathcal{F}}_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $\mathcal{F}$. Assuming that the $\mathcal{F}$-resolution dimension of every left $R$-module is finite and the class $\mathcal{F}$ is preserved by the coinduction functor ${\mathrm{Hom}}_R(A,-)$, we show that ${\mathcal{C}}_A$ is the class of all direct summands of left $A$-modules finitely (co)filtered by $A$-modules coinduced from $R$-modules from $\mathcal{C}$. Assuming that the class $\mathcal{F}$ is closed under countable products and preserved by the functor ${\mathrm{Hom}}_R(A,-)$, we prove that ${\mathcal{C}}_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $\mathcal{C}$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $\mathcal{F}$ have finite $\mathcal{F}$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $\omega +k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra (2010). In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz–Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.